Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Question
Chapter 7, Problem 7.35P
(a)
To determine
The projection operator
(b)
To determine
The eigenstates of
(c)
To determine
The “good” state and their energies to first order in the perturbation.
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Let a two-degree-of-freedom system be described by the Hamiltonian
1
H =(P +P)+ V(x, y)
and suppose the potential energy V is a homogenous function of degree -2:
V(xx, ày) = 2-2v(x,y) VÀ > 0.
Show that
$ = (xpy – yp.)² +2(x² + y²)V(x, y)
is a second constant of the motion independent of the Hamiltonian (Yoshida, 1987).
Therefore, this system is integrable.
The dynamics of a particle moving one-dimensionally in a potential V (x)
is governed by the Hamiltonian Ho = p²/2m + V (x), where p =
is the momentuin operator. Let E, n =
of Ho. Now consider a new Hamiltonian H
given parameter. Given A, m and E, find the eigenvalues of H.
-ih d/dx
1, 2, 3, ... , be the eigenvalues
Ho + Ap/m, where A is a
%3|
Problem 9.4
For the 2D LHO with K1 = K2 show that
and
[ê, ²] = 2ihxy,
(ê, p}] = -2ihxy
Problem 9.5
It follows from the above that
[ê., Ĥ] = 0
if K1 = K2 only
Work out the equivalent commutator for ê and é, with the
Hamiltonian. What do these mean?
Chapter 7 Solutions
Introduction To Quantum Mechanics
Ch. 7.1 - Prob. 7.1PCh. 7.1 - Prob. 7.2PCh. 7.1 - Prob. 7.3PCh. 7.1 - Prob. 7.4PCh. 7.1 - Prob. 7.5PCh. 7.1 - Prob. 7.6PCh. 7.2 - Prob. 7.8PCh. 7.2 - Prob. 7.9PCh. 7.2 - Prob. 7.10PCh. 7.2 - Prob. 7.11P
Ch. 7.2 - Prob. 7.12PCh. 7.2 - Prob. 7.13PCh. 7.3 - Prob. 7.15PCh. 7.3 - Prob. 7.16PCh. 7.3 - Prob. 7.17PCh. 7.3 - Prob. 7.18PCh. 7.3 - Prob. 7.19PCh. 7.3 - Prob. 7.20PCh. 7.3 - Prob. 7.21PCh. 7.3 - Prob. 7.22PCh. 7.4 - Prob. 7.23PCh. 7.4 - Prob. 7.24PCh. 7.4 - Prob. 7.25PCh. 7.4 - Prob. 7.26PCh. 7.4 - Prob. 7.27PCh. 7.4 - Prob. 7.28PCh. 7.4 - Prob. 7.29PCh. 7.5 - Prob. 7.31PCh. 7.5 - Prob. 7.32PCh. 7 - Prob. 7.33PCh. 7 - Prob. 7.34PCh. 7 - Prob. 7.35PCh. 7 - Prob. 7.36PCh. 7 - Prob. 7.37PCh. 7 - Prob. 7.38PCh. 7 - Prob. 7.39PCh. 7 - Prob. 7.40PCh. 7 - Prob. 7.42PCh. 7 - Prob. 7.43PCh. 7 - Prob. 7.44PCh. 7 - Prob. 7.45PCh. 7 - Prob. 7.46PCh. 7 - Prob. 7.47PCh. 7 - Prob. 7.49PCh. 7 - Prob. 7.50PCh. 7 - Prob. 7.51PCh. 7 - Prob. 7.52PCh. 7 - Prob. 7.54PCh. 7 - Prob. 7.56PCh. 7 - Prob. 7.57P
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